nLab
strong adjoint functor

Definition

For $C$ a cartesian closed category and $L,R:C\to C$ two endofunctors, they are called strong adjoints to each other if there is a natural isomorphism

$$[LX,A]\simeq [X,RA]$$[L X, A] \simeq [X, R A]

for all objects $X,A\in C$ and for $[-,-]$ the internal hom.

Properties

Notice that for $*$ the terminal object of $C$ we have that the global points of the internal hom give the external hom set

$$\Gamma [X,A]:=C(*,[X,A])\simeq C(X,A).$$\Gamma [X,A] := C(*, [X, A]) \simeq C(X,A) \,.

Therefore strongly adjoint functors are in particular adjoint functors in the ordinary sense.

References

For instance appendix 6 of

• Bill Lawvere, Outline of synthetic differential geometry (pdf)